as stated in Lemma REF . Such optimal estimates, for certain families of operators \(L\) ,
have already been investigated by the authors (and their collaborators) in earlier papers. For instance, besides the already mentioned reference [1]} for Hermite type operators, we have also considered a large class of Laguerre type operators \(L\)  in [2]}, while the Bessel operators (in the case \(\sigma =1/2\) ) were treated by I. Cardoso in [3]}.
